# Multivariate Interpolation Methods for Parametric Model Reduction

## DFG Project MIM4PMOR

### Project coordinator:

- Prof. Dr. Peter Benner

Max Planck Institute for Dynamics of Complex Technical Systems,

Computational Methods in Systems and Control Theory,

Tel: +49 (0)391-6110-451

E-mail: benner@mpi-magdeburg.mpg.de

### Researcher:

- Tobias Breiten

Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg,

Computational Methods in Systems and Control Theory,

Tel: +49 (0)391-6110-806

E-mail: breiten@mpi-magdeburg.mpg.de

### Project Description

Many computer-aided engineering problems involve analyzing large-scale parameter-dependent dynamical processes. Here, one can, e.g., think of optimization of geometry and topology in MEMS design. Model order reduction (MOR) techniques are often required to allow multiple simulations for varying parameter values in design studies or optimization algorithms. In case of parameters staying constant during one simulation cycle, there exist several generalizations of linear MOR methods like moment-matching, balanced truncation and rational interpolation, respectively. However, the situation becomes rather complicated if the parameters vary in time. Here, efficient reduction methods are still an open question.In this project, we investigate a beneficial connection between linear parameter-varying (LPV) control systems and so-called bilinear control systems. Although the latter ones formally belong to the class of nonlinear control systems, many linear reduction techniques have been shown to possess bilinear analogues. Moreover, embedding LPV systems in the class of bilnear control systems allows for studying parametric model reduction techniques that automatically take care of a desired structure preservation of the underyling parametric process.

### Duration and Funding

- January 2010 - August 2010: Chemnitz University of Technology
- since September 2010: MPI Magdeburg, funded by DFG (since 2012)

### Related Publications

@Article{BenM12b, author = {Peter Benner and Thomas Mach}, title = {Computing All or Some Eigenvalues of Symmetric $\mathcal{H}_{\ell}$-Matrices}, publisher = {SIAM}, year = {2012}, journal = {SIAM Journal on Scientific Computing}, volume = {34}, number = {1}, pages = {A485-A496}, url = {http://epubs.siam.org/doi/abs/10.1137/110836742}, doi = {10.1137/110836742} } @TECHREPORT{MPIMD11-02, author = {Benner, Peter; Breiten, Tobias}, title = {Interpolation-Based ${\cal H}_2$-Model Reduction of Bilinear Control Systems}, institution = {Max Planck Institute Magdeburg Preprints}, year = 2011, number = {MPIMD/11-02}, month = {June} 859–885 } | Interpolation-Based ℋ₂-Model Reduction of Bilinear Control Systems; Benner, Peter; Breiten, Tobias; SIAM Journal on Matrix Analysis and Applications : Vol. 33, No. 3, 859–885;2012. doi: 10.1137/110836742 also available as preprint MPIMD/11-02. |

@INPROCEEDINGS{BenB11, author = {P. Benner and T. Breiten}, title = {On {$\mathcal{H}_2$}-model reduction of linear parameter-varying systems}, journal = {PAMM}, volume = {11}, number = {1}, publisher = {WILEY-VCH Verlag}, url = {http://dx.doi.org/10.1002/pamm.201110391}, doi = {10.1002/pamm.201110391}, pages = {805--806}, year = {2011}, } | On ℋ₂-model reduction of linear parameter-varying systems; Benner, Peter; Breiten, Tobias; Proceedings in Applied Mathematics and Mechanics : Volume 11, Issue 1, pages 805–806, December 2011;Wiley InterScience; 2011. doi: 10.1002/pamm.201110391. |

@ARTICLE{BenBD11, author = {P. Benner and T. Breiten and T. Damm}, title = {Generalised tangential interpolation for model reduction of discrete-time MIMO bilinear systems}, journal = {Internat. J. Control}, year ={2011}, volume = {84}, pages = {1398--1407}, number = {8}, doi = {http://dx.doi.org/10.1080/00207179.2011.601761}, } | Generalised tangential interpolation for model reduction of discrete-time MIMO bilinear systems; Benner, Peter; Breiten, Tobias; Damm, Tobias; International Journal of Control : Vol. 84 Issue 8, pp. 1398-1407;2011. doi: 10.1080/00207179.2011.601761. |

@INPROCEEDINGS{BenBD10, author = {P. Benner and T. Breiten and T. Damm}, title = {Krylov Subspace Methods for Model Order Reduction of Bilinear Discrete-Time Control Systems}, journal = {PAMM}, volume = {10}, number = {1}, publisher = {WILEY-VCH Verlag}, url = {http://dx.doi.org/10.1002/pamm.201010293}, doi = {10.1002/pamm.201010293}, pages = {601--602}, year = {2010}, } | Krylov Subspace Methods for Model Order Reduction of Bilinear Discrete-Time Control Systems; Benner, Peter; Breiten, Tobias; Damm, Tobias; Proceedings in Applied Mathematics and Mechanics : Vol. 10 Issue 1, pp. 601-602;Wiley InterScience; 2010. doi: 10.1002/pamm.201010293. |

@ARTICLE{BreD10, author = {T. Breiten and T. Damm}, title = {Krylov subspace methods for model order reduction of bilinear control systems}, journal = {Syst. Control Lett.}, year = {2010}, volume = {59}, pages = {443--450}, doi = {http://dx.doi.org/10.1016/j.sysconle.2010.06.003}, } | Krylov subspace methods for model order reduction of bilinear control systems; Breiten, Tobias; Damm, Tobias; Systems & Control Letters : Vol. 59 Issue 8, pp. 443-450 ;2010. doi: 10.1016/j.sysconle.2010.06.003. |